THE CONCEPT OF STATISTICAL SIGNIFICANCE: CHI-SQUARE AND THE NULL HYPOTHESIS

READINGS Pollock, Essentials, ch. 5 and ch. 6, pp. 121-135 Pollock, SPSS Companion, ch. 7

OUTLINE Strategies for Sampling Establishing Confidence Intervals Chi-Square and the Null Hypothesis Critical Values of Chi-Square

Why Sample? Goal: description of a population Advantages: savings of time and money Basic paradox: credibility of results from a sample depends on size and quality of the sample itself, and not on the size of the population

Types of Samples Probability sampling: Every individual in the population has a known probability of being included in the sample Random sample (SRS): each individual has an equal chance of being selected, and all combinations are equally possible Systematic sample: every kth individual—more or less equivalent to SRS if first selection is made through random process Stratified sample: individuals separated into categories, and independent (SRS) samples selected within the categories Cluster sample: population divided into clusters, and random sample (SRS) then drawn of the clusters

Parameters and Statistics A parameter is a number that describes the population. It is a fixed number, though we do not know its value. A statistic is a number that describes a sample. We use statistics to estimate unknown parameters. A goal of statistics: To estimate the probability that the null hypothesis holds true for the population. Forms: Parameter may not fall within a confidence band that can be placed around a sample statistic, or A relationship observed within a sample may not have a satisfactory probability of existing within the population.

Problems with Sampling (I) Bias: A consistent, repeated deviation of the sample statistic from the population parameter Convenience sampling Voluntary response sampling Solution: Use SRS Variation: Signal: large standard deviation within sample Range of sample statistics Solution: Use larger N

Problems in Sampling (II) Ho for Sample Accepted Rejected Ho for Population True Type I False Type II Where Ho = null hypothesis

What is Chi-square? A measure of “significance” for cross-tabular relationships Where fo = “observed frequency” (or cell count) And fe = “expected frequency” (or cell count) X2 = Σ (fo – fe)2/fe

Calculating Expected Frequencies: fe = col Σ (row Σ/total N) for upper left-hand cell = 802 (200/1,679) = 95.5 fo = 44 fo – fe = 44 – 95.5 = -51.5 (fo – fe)2 = 2,652.25 (fo – fe)2/fe = 27.77

Conceptualizing Chi-Square Expected frequencies represent the “null hypothesis” (no relationship) Observed frequencies present visible results Question 1: Are observed frequencies different from expected frequencies? Question 2: Are they sufficiently different to allow us to reject the possibility that the true relationship (within the universe of case) is null?

Figuring Degrees of Freedom: df = (r – 1)(c – 1) Illustration: Given marginal values, ________X________ __Y__ L H Σ L 30 50 H 50 Σ 60 40 100 and df = 1

Characteristics of Chi-Square Distribution for null hypothesis has a known distribution—skewed to the right Specific distributions have corresponding degrees of freedom, defined as (r-1)(c-1) For a 2x2 table, chi-square of 3.841 or greater would occur no more than 5% of the time in event of null hypothesis (thus, “.05 level or better”)

POSTSCRIPT X2 = f (strength of relationship, sample size) The stronger the observed relationship within the sample, the higher the X2 The larger the sample (SRS), the higher the X2 The higher the X2 (given degrees of freedom), the greater the probability that null hypothesis does not hold in the population (p < .05)

Limitations of Chi-Square No more than 20% of expected frequencies less than 5 and all individual expected frequencies are 1 or greater Directly proportional to N observations Rejection of null hypothesis does not directly confirm strength or direction of relationship

Review: Summary Measures for Cross-Tabulations Lambda-b PRE, ranges from zero to unity; measures strength only Gamma Form and strength (-1 to +1), based on “pairs” of observations Chi-square Significance, based on deviation from “null hypothesis”